Simplify the following expression and state the condition under which the simplification is valid: $k = \dfrac{x^2 + 7x - 18}{x^2 - 12x + 20}$
First factor the expressions in the numerator and denominator. $ \dfrac{x^2 + 7x - 18}{x^2 - 12x + 20} = \dfrac{(x + 9)(x - 2)}{(x - 10)(x - 2)} $ Notice that the term $(x - 2)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(x - 2)$ gives: $k = \dfrac{x + 9}{x - 10}$ Since we divided by $(x - 2)$, $x \neq 2$. $k = \dfrac{x + 9}{x - 10}; \space x \neq 2$